According to Davis: Connecting principles and practices

In this article, the author allows Robert B. Davis to state for himself his own Principles concerning how children learn, and how teachers can best teach them. These principles are put forward in Davis’ own words along with detailed documentation. The author goes on compare Davis’ words with his practices. A single Davis video (Towers of Hanoi) is analyzed to determine if, and to what extent, his principles are evident in his teaching of this lesson.     

Infinitely small quantities in Cauchy's textbooks

The fundamental role of infinitely small quantities for his teaching of the calculus was underlined by Cauchy himself in the introduction to his Cours d'analyse of 1821 and in the announcements of his later textbooks. First steps toward theories of such quantities which are briefly denoted as variables having zero as their limit were made by Cauchy, who represented them by sequences converging to zero (in the Cours) or by functions vanishing at zero (since 1823). It is shown that the famous so-called errors of Cauchy are correct theorems when interpreted with his own concepts. A few gaps in his proofs are explained by the hypothesis that he tacitly assumed continuity. No assumptions on uniformity or on nonstandard numbers are needed. Finally, some possible completions of Cauchy's rudimentary theories of infinitesimals are ventured.     

Normed rings generated by generalized convolutions

Normed rings are considered that are generated by the generalized shift operation that occurs in the study of the analytic properties of Urbanik algebras. Possible applications are illustrated by the example of a counterpart of the classical criterion of positive definiteness and by an inversion formula for generalized characteristic functions of Urbanik.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 12–18, 1980.The author expresses his sincere gratitude to A. A. Zinger for his interest.     

Cooperative processing of information via choice at an information set

A specific structure is added to the model of Bayesian society, and the associated (more involved) version of the Bayesian incentive compatible strong equilibrium concept is proposed. The equilibrium endogenously explains whether or not playerj in coalitionS, in pursuit of his self-interest, decides to pass on his private information to the other members ofS, and if he does, which part of his private information he decides to pass on. Generic existence theorems for this equilibrium are established.     

A two-stage stochastic model for labour wastage

A stochastic model for labour wastage is presented which accounts for the employee's variability on the propensity to leave through dependence on both his length of service and his tenure-in-current-state. The basic assumption underlying the model is that the employee's personal characteristics, the job characteristics and the external labour market conditions are stochastic over time, therefore affecting his decision to stay or leave the company. It is shown that the model provides a good fit to a variety of observed leaving patterns for several companies reported in the literature, explains the relationships among a number of important occupational variables, and is useful for planning purposes in predicting future developments.     

Properties of subspaces of the space of affine functions

Conditions are presented for weak (strong) convergence, in the entire space A(x), of a sequence of positive operators to a fixed operator S on the basis of its weak (strong) convergence to S in a subspace. The criterion of Bauer's simplex is presented.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 307–318, February, 1975.In conclusion the author expresses his deep gratitude to Yu. A. Shashkin for his many valuable remarks.     

Middle dimension homologies of a codimension 2 submanifold. II

This paper is devoted to the estimates for the middle dimension homologies of codimension 2 submanifoldA of a smooth manifold X. Proofs are based on the information about homologies of the ramified covering of X branched alongA. This information is extracted from the Atiyah-Singer signature formulas and from some corollaries of the Smith theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 182–185, 1976.The author takes this oppotunity to express his gratitude to V. A. Rokhlin for the formulation of the problem and for his interest in this paper.     

E.W. Chittenden and the early history of general topology

E.W. Chittenden's work and its influence on the early history of general topology are examined. Particular attention is given to his work in metrization theory and its role in the background of the Aleksandrov-Uryson Metrization Theorem. A recounting of Professor Chittenden's career, a list of his students and his publications and a chronology in the early history of General Topology are also included.     

On the discrete spectrum of a perturbed periodic Schrödinger operator

A perturbed periodic Schrödinger operator is considered. Conditions on the perturbation are obtained, under which the discrete spectrum in the gap is finite or, on the contrary, is infinite. In the latter case there are given formulas for the asymptotic behavior of the eigenvalues with respect to the index and conditions under which these formulas hold. The results are formulated in terms of a model problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 157–162, 1991.In conclusion the author expresses his gratitude to his scientific adviser M. Sh. Birman.     

Hamiltonian structure of some two-component systems

Hamiltonian structures are found for a number of two-component nonlinear evolution systems having higher conservation laws, and their integrability is proved by the MISP.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 143–153, 1986.The author considers it his pleasant duty to express profound gratitude to P. P. Kulish for his constant interest in the work.     

Spectral properties of a linearized magnetohydrodynamics model

One considers the linear oscillations of a perfectly conducting nonviscous plasma in an external rectilinear magnetic field. One studies the frequency spectrum corresponding to a fixed harmonic mode. One has investigated the asymptotic behavior of the discrete spectrum at infinity and around the boundary of the essential spectrum.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 120–123, 1985.The author expresses his deep gratitude to his scientific adviser Professor M. Sh. Birman for his help and for numerous discussions and to A. E. Lifshits for consultations and for his interest in this paper.     

Hyperfoliations on compact 3-manifolds with restrictions on the external curvature of leaves

C∞-foliations of codimension 1 on compact Riemannian 3-manifolds are studied. New classes of foliations, namely hyperbolic, elliptic, and parabolic foliations, are considered. Examples of such foliations are presented. In particular, aC∞-metric of nonnegative sectional curvature onS ³ such that the Reeb foliation is parabolic with respect to this metric is constructed. Analytic 3-manifolds with sectional curvature of constant sign admitting parabolic foliations are classified. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 651–659, May, 1998. The author wishes to express his thanks to Professor A. A. Borisenko for his supervision, and to Yu. A. Nikolaevskii for useful advice in the process of preparing the present paper.     

The optimal choice of initial approximations to solutions of regularized equations in the theory of branching

In [1, 2], V. A. Trenogin and the author proposed a method for constructing regularizing equations (R.E.) for ill-posed problems in the theory of branching [3, 4]. Here we consider the problem of choosing initial approximations, that are optimal in a certain sense, to simple solutions of R.E. in a neighborhood of a branching point.Translated from Matematicheskie Zametki, Vol. 20, No. 2, pp. 273–278, August, 1976.The author thanks V. A. Trenogin for his interest in this research and his valuable advice.     

盖尔范德在整值整函数理论上的工作

本文基于原始文献,利用历史分析和比较的方法,首次研究了盖尔范德在整值整函数理论方面的工作及影响.他的工作奠定了该理论发展的重要方向.文章分析了其工作背景;研究了他的有关工作,揭示了其思想的演变过程;探讨了其工作的重要影响.     

A lower bound for linear forms in values of certain hypergeometric functions

Lower bounds are found for linear forms in values of certain hypergeometric functions and their derivatives, analogous to the bounds obtained by K. Mahler for the linear form of the numbers 1, e, e²,..., e^m.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 19–28, July, 1970.We wish to thank A. B. Shidlovskii for his interest in this work and for his many useful suggestions.     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

Hume’s Argument for the Ontological Independence of Simple Properties

In this paper, I will reconstruct Hume's argument for the ontological (in the sense of rigid existential) independence of simple properties in A Treatise of Human Nature, Book 1 (1739). According to my reconstruction, the main premises of the argument are the real distinctness of every perception of a simple property, Hume's Separability Principle and his Conceivability Principle. In my view, Hume grounds the real distinctness of every perception of a simple property in his atomistic theory of sense perception and his Copy Principle. I will also show why Hume's argument should be seen as relevant nowadays. David Lewis and his followers in metaphysics continue Hume's line of thinking in this respect, which is opposed by power ontologists (Brian Ellis, Stephen Mumford), for example.     

The representation of integral functions by series of the form ∑ n=1 ∞ α n fff(λ n z)

The necessary and sufficient conditions that an integral function, whose order satisfies certain conditions, should be represented by a series of integral functions of the formf(_nz) are indicated.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 579–588, November, 1968.The author wishes to express his deep gratitude to his scientific director A. F. Leont'ev for his attention during the preparation of the paper.     

Stability of the characterization of the multivariate normal distribution in the Skitovich-Darmois theorem

A numerical estimate of stability is obtained in a multidimensional variant of the Skitovich-Darmois theorem when the original random vectors, as well as linear combinations of them, are assumed to be -independent.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 61, pp. 5–16, 1976.The author expresses his profound thanks to V. M. Zolotarev for his continuing attention to the work.     

A new approach to constrained function optimization

A new approach to the constrained function optimization problem is presented. It is shown that the ordinary Lagrange multiplier method and the penalty function method may be generalized and combined, and the new concept ofmultiplier function is introduced. The problem may then be converted into an unconstrained well-conditioned optimization problem. Methods for numerical solution are discussed, and new algorithms are derived.The author wishes to express his gratitude to Professor K. J. Åström for his encouragement and assistance and to Professor P. Falb for valuable suggested improvements. This work was supported by the Swedish Board for Technical Development, Contract No. 70-337/U270.